density of dark matter halo

Question

I was surfing the net for knowing the actual density of dark matter halo for a particular M_h(Mass of the halo) and finally, I got two equations:

for pseudo-isothermal halo

and

by NFW-profile

So, with which density distribution of dark matter halo should I go corresponding to any M_h?

Answer 1

tl;dr The answer is opinion-based, but choosing an NFW profile probably won't offend anyone.

Pseudo-isothermal profile

There are many more or less physically motivated functional forms for the density profile of a dark matter halo. The pseudo-isothermal sphere is a less unphysical profile than the simple, isothermal profile (which is unphysical in the center and has a diverging integrated mass for $r\rightarrow\infty$). It gives good fit to the rotation curves in the optical disk of galaxies, but not to the light distribution.

NFW profile

Arguably, the most popular profile is the NFW profile (Navarro, Frenk, & White 1996;1997) — with a total of ~13,000 citations, these two papers are among the most cited papers in astrophysics ever. Its popularity is probably partly due to its simplicity. It isn't really physically motivated, but simply turned out to give good fits to numerical simulations.

Einasto profile

However, subsequent N-body simulations with higher resolution ( Navarro et al. 2004; Hayashi & White 2008; Gao et al. 2008; Springel et al. 2008 ) showed small, but systematic, deviations from the NFW profile which are more accurately modeled with an Einasto (1965) profile: $$ \rho(r)=\rho_{-2} \exp \left[\frac{-2}{\alpha}\left\{\left(\frac{r}{r_{-2}}\right)^{\alpha}-1\right\}\right], $$ where $r_{-2}$ is the radius at which the slope of the profile is $-2$, $\rho_{-2} = \rho(r_{-2})$, and $\alpha$ is a fitting parameter with best-fit values that typically lie in the range $[0.12,0.25]$, increasing with halo mass.

However, since there isn't much physics here either, and this form is more complicated, the NFW profile seems to remain the profile of choice for most people.

Other profiles

Many profiles can be written as double power-laws $$ \rho(r)=\rho_{0}\left(\frac{r}{r_{0}}\right)^{-\gamma}\left[1+\left(\frac{r}{r_{0}}\right)^{\alpha}\right]^{(\gamma-\beta) / \alpha}, $$ where the parameters $\alpha$, $\beta$, and $\gamma$ can take a variety of values (Tab. 5.1 from Mo et al. 2010): $$ \begin{array}{lll} \hline \hline (\alpha, \beta, \gamma) & \text{Name} & \text{Reference} \\ \hline(1,3,1) & \text{NFW profile} & \text{Navarro et al. (1997)} \\ (1,4, \gamma) & \text{Dehnen profile} & \text{Dehnen (1993)} \\ (1,4,1) & \text{Hernquist profile} & \text{Hernquist (1990)} \\ (1,4,2) & \text{Jaffe profile} & \text{Jaffe (1983)} \\ (2,2,0) & \text{Modified isothermal sphere} & \text{Sackett & Sparke (1990)} \\ (2,3,0) & \text{Modified Hubble profile} & \text{Binney & Tremaine (1987) }\\ (2,4,0) & \text{Perfect sphere} & \text{de Zeeuw (1985)} \\ (2,5,0) & \text{Plummer sphere} & \text{Plummer (1911)} \\ \hline \hline \end{array} $$ At small and large radii $\rho$ is proportional to $r-\gamma$ and $r-\beta$, respectively, while $\alpha$ gives the sharpness of the break.

Some profiles are better at describing the inner structure, some are better farther out, some are better when you include gas, etc. What you choose depends on your particular problem, and your mood.